Advanced Algebra Lesson Objectives DAA

Advanced Algebra Lesson Objectives

Semester 1 Semester 2

Chapter 0
In this chapter you will:

• solve problems both on your own and as a group
• use pictures and graphs as problem-solving tools
• learn a four-step process for solving problems with symbolic algebra
• practice strategies for organizing information before you solve a problem.

0.1The students will:

• Learn and review multiple ways to solve problems.
  • Introduce a 4-step outline for solving problems
  • Use pictures, diagrams, and graphs as problem-solving tools
  • Find the slope of a line from a graph
  • Find the slope of a line from a pair of points
  • Learn to work in a cooperative group

0.2 The students will:

• Learn and review symbolic representation in algebra.
  • Translate English phrases into algebraic symbols assigning a variable to each unknown quantity
  • Encounter absolute-value notation
  • Review solving equations and solving systems of equations

0.3 The students will:

• Learn good ways to organize information
  • Practice using dimensional analysis and unit conversion
  • Solve logic problems
  • Improve at working cooperatively

Chapter 2
In this chapter you will:

• create, interpret, and compare graphs of data sets
• calculate numerical measures that help you understand and interpret a data set
• make conclusions about a data set and compare it with other data sets based on graphs and numerical values.

2.1 The students will

• Learn about measures of central tendency and box plots.
  • Review mean, median, and mode
  • Learn sigma notation for summation
  • Create box plots from 5-number summaries
  • Create 5-number summaries from box plots
  • Describe the shape and spread of a data set from box plots
  • Use box plots to compare the centers, shapes, and spreads of data sets
  • Construct a data set having various statistics

2.2 The students will

• Learn about measures of spread
  • Develop a concept of spread (variability)
  • Use a calculator to find standard deviation
  • Derive formulas for standard deviation and variance - find standard deviation by hand
  • Understand deviation from the mean
  • Distinguish measures of spread relative to the mean from measures of spread relative to the median
  • Understand and apply definitions of outlier

2.3 The students will

• Learn about histograms and percentile ranks
  • Distinguish bar graphs from histograms
  • Find the approximate number of data items, the range, and the median by studying a histogram
  • Understand the effect of different bin widths in histograms
  • Create box plots and histograms in order to analyze specific characteristics of a data set
  • Determine the percentile rank of a data item either from raw data or from a histogram
  • Connect percentile rank and standard deviation

Chapter 1
In this chapter you will:

• recognize and visualize mathematical patterns called sequences
• write recursive definitions for sequences
• display sequences with graphs
• investigate what happens to sequences in the long run.

1.1 The students will

• Learn about recursively defined sequences
  • Introduce recursive formulas for sequences
  • Explore arithmetic sequences
  • Practice using recursive notation
  • Define and use geometric sequences
  • Use recursively defined sequences to model applications

1.2 The students will

• Learn about modeling growth and decay
  • Use geometric sequences to model decay
  • Use geometric sequences to model growth

1.3 The students will

• Learn about limits
  • Explore long run values
  • Begin to understand the concept of limit
  • Investigate shifted geometric sequences

1.4 The students will

• Learn about graphing sequences
  • Recognize arithmetic and geometric sequences from their graphs
  • Use graphs to check whether a recursive formula is a good model for the data

1.5 The students will

• Learn about loans and investments
  • Use a recursive formula to model a loan
  • Use a recursive formula to model an investment

Chapter 3
In this chapter you will:

• review linear equations in intercept form and point-slope form
• explore connections between arithmetic sequences and linear equations
• find lines of fit for data sets that are approximately linear
• solve systems of linear equations.

3.1 The students will

• Learn about linear equations and arithmetic sequences
  • Given a recursive formula, find n for a given un
  • Graph an arithmetic sequence to locate the intercept and determine the slope
  • Recognize slope as the common difference in an arithmetic sequence;
  • Use the intercept and slope to write a linear equation in x and y
  • Recognize that an arithmetic sequence is always linear

3.2 The students will

• Learn about slope
  • Use recursion in application contexts
  • Define domain and range

3.3 The students will

• Learn about fitting a line to data
  • Find a line of fit for data that are approximately linear
  • Use interpolation and extrapolation

3.4 SKIP
The students will

• Learn about the median-median line
  • Practice writing equations for lines through two points
  • Review how to determine the equation of a line parallel to another line
  • Introduce median-median line
  • Compare median-median line to other lines of fit

3.5 SKIP
The students will

• Learn about residuals
  • Define residual, root mean square error
  • Calculate the residual sum from a small number of data points
  • Use residuals to test the fit of a line
  • Calculate the root mean square error for a small number of data points
  • Use the root mean square error to find the error range in the context of a problem

3.6 The students will

• Learn about linear systems
  • Examine problems involving multiple conditions that must be satisfied simultaneously
  • Understand the visual representation of a solution to a system of equations
  • Solve systems of equations graphically, using the transitive property of equality, and using a table

3.7 The students will

• Learn about substitution and elimination
  • Solve systems of equations using substitution
  • Explore how the addition and multiplication properties of equations can be used to solve systems of equations by elimination

Chapter 4
In this chapter you will:

• interpret graphs of functions and relations
• review function notation
• learn about the linear, quadratic, square root, absolute-value, and semicircle families of functions
• apply transformations - translations, reflections, stretches, and shrinks - to the graphs of functions and relations
• transform functions to model real-world data

4.1 The students will

• Learn about interpreting graphs
  • Identify independent and dependent variables
  • Interpret features of a qualitative graph, including rates of change and x- and y-intercepts
  • Decide whether a graph (or a function) is discrete or continuous given a description of the variables
  • Draw a qualitative graph from a context scenario and create a context scenario given a qualitative graph
  • Distinguish between linear change and non-linear change.

4.2 The students will

• Learn about function notation
  • Define function as "a relation with at most one y-value for any x-value"
  • Review function notation
  • Review the vertical line test for functions
  • Distinguish between functions and relations
  • Define the domain and range of a function.

4.3 The students will

• Learn about lines in motion
  • Review linear equations
  • Describe translations of a line in terms of horizontal and vertical shifts
  • Write the equation of a translated line using h and k
  • Understand point-slope form as a translation of the line with its equation written in intercept form
  • Apply translations to functions.

4.4 The students will

• Learn about translations and the quadratic family
  • Define the parent quadratic function y = x2
  • Determine elements of equations that produce translations of the graphs of parent functions (h and k)
  • Introduce the (non-stretched) vertex form of the graph of a parabola, y = (x - h)2 + k
  • Define parabola, vertex of a parabola, and line of symmetry.

4.5 The students will

• Learn about reflections and the square root family
  • Define reflection
  • Define the parent square root function, y = x1/2
  • Define the square root symbol and function as the positive root
  • Compare f(x), -f(x), f(-x), and -f(-x); Apply the square root function in context
  • Apply reflections to functions in general
  • Symbolically solve the equation a + (x + b)1/2 = c for x.

4.6 The students will

• Learn about stretches and shrinks and the absolute value family
  • Define absolute value and its notation, and use it to model distance
  • Define the parent absolute-value function, y = |x|, and the absolute-value family, y = a|x-h| + k
  • Calculate horizontal and vertical stretch or shrink factors from points on the image of a graph
  • Apply horizontal and vertical stretches and shrinks to functions in general.

4.7 The students will

• Learn about transformations and the circle family
  • Define unit circle and derive the equation x
  • Express a circle as two semicircle functions
  • Define ellipse as "a vertical and/or horizontal dilation of a circle"
  • Transform a circle to get an ellipse
  • Apply transformations to relations and to a new function expressed in terms of f(x)
  • Summarize transformations - translations, reflections, rotations, and stretches and shrinks.

4.8 The students will

• Learn about compositions of functions
  • Define composition of functions and learn the notation
  • See transformations of two or three steps as the composition of functions
  • Apply composition to real world contexts
  • Distinguish composition from the product of functions
  • Understand composition both graphically and numerically.

Chapter 12
In this chapter you will:

• learn about randomness and the definition of probability
• count numbers of possibilities to determine probabilities
• determine expected values of random variables

12.1 The students will:

• Learn about randomness and probability.
  • Define experimental and theoretical probability
  • Simulate experimental probability on a calculator
  • Define and calculate geometric probability.

12.2 The students will:

• Learn about counting outcomes and tree diagrams.
  • Use tree diagrams as an aid to counting possibilities for compound events
  • Use the multiplication rule for independent events
  • Explore conditional probability.

12.3 The students will

• Learn about mutually exclusive events and Venn diagrams
  • Explore mutually exclusive events
  • Use Venn diagrams as a tool for breaking down compound events into mutually exclusive events
  • Understand the addition rule for finding the probabilities of events described by a Venn diagram
  • Differentiate between mutually exclusive events and independent events.